What is the ruling on someone who divorced his wife with an oath of divorce, then said to her, "By divorce, I will not take you back," and repeated it while in a state of extreme anger, then had intercourse with her? Is the divorce considered to have taken place in this case, and what is the expiation for this oath?

If the immediate divorce oath (yamin al-talaq al-munjaz) is common in your country, then the divorce is binding. And since he swore an oath of divorce that he would not take you back, divorce befalls him upon his taking you back. This falls under the category of conditional divorce (talaq mu'allaq) according to the majority of scholars. So if your husband takes you back through intercourse after the first divorce, then two divorces are counted against him. He has the right to take you back before the expiration of your 'idda (waiting period), provided he had not divorced you before that. The return (raj'ah) occurs by word or action with intention, and by intercourse even without intention, according to the sounder opinion. In summary: If your husband divorced you immediately (talaq munjaz), then swore by divorce that he would not take you back, and then took you back through intercourse, a second divorce is counted against him according to the majority of scholars. As for the opinion of Ibn Taymiyyah, he held that swearing by an oath of divorce only necessitates an expiation for an oath (kaffarat yamin). It is preferable to present the matter to the Sharia courts or knowledgeable scholars.